View source for Stokes' theorem

{{short description|Theorem in vector calculus}}
{{Calculus |Vector}}
[[Image:Stokes' Theorem.svg|thumb|right|An illustration of Stokes' theorem, with surface {{math|Σ}}, its boundary {{math|∂Σ}} and the normal vector {{mvar|n}}. The direction of positive circulation of the bounding contour {{math|∂Σ}}, and the direction {{mvar|n}} of positive flux through the surface {{math|Σ}}, are related by a [[Right-hand rule|right-hand-rule]] (i.e., the right hand the fingers circulate along {{math|∂Σ}} and the thumb is directed along {{mvar|n}}).]]

'''Stokes' theorem''',<ref>{{Cite book |last=Stewart |first=James |author-link=James Stewart (mathematician) |url=https://patemath.weebly.com/uploads/5/2/5/8/52589185/james-stewart-calculus-early-transcendentals-7th-edition-2012-1-20ng7to-1ck11on.pdf |title=Calculus – Early Transcendentals |publisher=[[Brooks/Cole]] |year=2012 |isbn=978-0-538-49790-9 |edition=7th |pages=1122}}</ref> also known as the '''Kelvin–Stokes theorem'''<ref name="iwahori">{{Cite book |last=Nagayoshi |first=Iwahori |author-link=Nagayoshi Iwahori |url=http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1039-4.htm |title=微分積分学 (Bibun sekibungaku) |publisher=[[:ja:裳華房|Shokabo]] |year=1983 |isbn=978-4-7853-1039-4 |language=ja |oclc=673475347}}</ref><ref name="fujimno">{{Cite book |last=Atsuo |first=Fujimoto |title=現代数学レクチャーズ. C 1, ベクトル解析 (Gendai sūgaku rekuchāzu. C(1), Bekutoru kaiseki) |publisher=[[:ja:培風館|Baifukan]] |year=1979 |language=ja |oclc=674186011}}</ref> after [[Lord Kelvin]] and [[Sir George Stokes, 1st Baronet|George Stokes]], the '''fundamental theorem for curls''', or simply the '''curl theorem''',<ref>{{Cite book |last=Griffiths |first=David J. |author-link=David J. Griffiths |url= |title=Introduction to Electrodynamics |title-link=Introduction to Electrodynamics |publisher=Pearson |year=2013 |isbn=978-0-321-85656-2 |edition=4th |series= |location= |pages=34}}</ref> is a [[theorem]] in [[vector calculus]] on <math>\R^3</math>. Given a [[vector field]], the theorem relates the [[Surface integral|integral]] of the [[Curl (mathematics)|curl]] of the vector field over some surface, to the [[line integral]] of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence:
: The [[line integral]] of a vector field over a loop is equal to the [[surface integral]] of its ''[[Curl (mathematics)|curl]]'' over the enclosed surface.

Stokes' theorem is a special case of the [[generalized Stokes theorem]].<ref name="DTPO">{{Cite book |last=Conlon |first=Lawrence |url={{Google books|plainurl=yes|id=r2K31Pz5EGcC|page=194 }} |title=Differentiable manifolds |publisher=[[Birkhäuser]] |year=2008 |isbn=978-0-8176-4766-7 |edition=2. |series=Modern Birkhäuser classics |location=Boston; Berlin}}</ref><ref name="lee">{{Cite book |last=Lee |first=John M. |url={{Google books|plainurl=yes|id=xygVcKGPsNwC|page=421 }} |title=Introduction to smooth manifolds |publisher=[[Springer Publishing|Springer]] |year=2012 |isbn=978-1-4419-9982-5 |edition=2nd |series=Graduate texts in mathematics |volume=218 |location=New York; London}}</ref> In particular, a vector field on <math>\R^3</math> can be considered as a [[differential form|1-form]] in which case its curl is its [[exterior derivative]], a 2-form.

== Theorem ==

Let <math>\Sigma</math> be a smooth oriented surface in <math>\R^3</math> with boundary <math>\partial \Sigma \equiv \Gamma </math>. If a vector field <math>\mathbf{F}(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z))</math> is defined and has continuous first order [[partial derivatives]] in a region containing <math>\Sigma</math>, then
<math display="block">
\iint_\Sigma (\nabla \times \mathbf{F}) \cdot \mathrm{d} \mathbf{\Sigma}  =  \oint_{\partial\Sigma} \mathbf{F} \cdot 
\mathrm{d}\mathbf{\Gamma}.
</math>
More explicitly, with <math>\wedge</math> being the [[wedge product]], the equality says that
<math display="block">
\begin{align}
&\iint_\Sigma \left(\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \right)\,\mathrm{d}y\wedge \mathrm{d}z +\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\, \mathrm{d}z\wedge \mathrm{d}x  +\left (\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\, \mathrm{d}x\wedge \mathrm{d}y\right) \\
& = \oint_{\partial\Sigma} \Bigl(F_x\, \mathrm{d}x+F_y\, \mathrm{d}y+F_z\, \mathrm{d}z\Bigr).
\end{align}
</math>

The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary.  Surfaces such as the [[Koch snowflake]], for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in [[Lebesgue integration|Lebesgue theory]] cannot be defined for a non-[[Lipschitz function|Lipschitz]] surface.  One (advanced) technique is to pass to a [[weak formulation]] and then apply the machinery of [[geometric measure theory]]; for that approach see the [[coarea formula]].  In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of <math>\R^2</math>.

A more detailed statement will be given for subsequent discussions. 
Let <math>\gamma:[a,b]\to\R^2</math> be a [[piecewise]] smooth [[Jordan curve|Jordan plane curve]]: a simple closed curve in the plane. The [[Jordan curve theorem]] implies that <math>\gamma</math> divides <math>\R^2</math> into two components, a [[compact space|compact]] one and another that is non-compact. Let <math>D</math> denote the compact part; then <math>D</math> is bounded by <math>\gamma</math>.  It now suffices to transfer this notion of boundary along a continuous map to our surface in <math>\R^3</math>.  But we already have such a map: the [[Parametrization (geometry)|parametrization]] of <math>\Sigma</math>.

Suppose <math>\psi:D\to\R^3</math> is [[piecewise]] smooth at the [[Neighbourhood (mathematics)|neighborhood]] of <math>D</math><ref group="note" name=NeighbourhoodofD>From the definition of <math>D</math>, <math>D</math> is obviously a bounded closed set in <math>\R^2</math>. "A neighborhood of D" means "an open set in <math>\R^2 </math> that contains D."
</ref>　, with <math>\Sigma=\psi(D)</math>.<ref group="note" name=imgg><math>\Sigma=\psi(D)</math> represents the [[Image (mathematics)|image set]] of <math>D</math> by <math>\psi</math></ref> If <math>\Gamma</math> is the [[space curve]] defined by <math>\Gamma(t)=\psi(\gamma(t))</math><ref group="note" name=cgamma><math>\Gamma</math> may not be a [[Jordan curve]] if the loop <math>\gamma</math> interacts poorly with <math>\psi</math>.  Nonetheless, <math>\Gamma</math> is always a [[loop (topology)|loop]] or loops, and topologically a [[connected sum]] of [[countable set|countably many]] Jordan curves, so that the integrals are well-defined. </ref> then we call <math>\Gamma</math> the boundary of <math>\Sigma</math>, written <math>\partial\Sigma</math>
<ref group="note" name=boundery>Even if we consider two-dimensional polar coordinates, then if we let ∂Σ = Γ, then ∂Σ can clearly contain a topological interior point of Σ. However, in [[orientable]] [[combinatorial manifolds]], this is not such a serious problem, since the line integrals over the interior points cancel out: in an orientable manifold, the line integral over ∂Σ coincides with the line integral over the true boundary.An even more important example is that this theorem also applies to manifolds without a topological boundary, such as the sphere or the torus. In such cases, if ∂Σ = Γ, then ∂Σ is completely contained within Σ. However, in such cases, the resulting line integrals over Γ completely cancel each other out. This allows us to assert that '''the surface integral of the curl of a vector field on a manifold without a boundary is zero'''.</ref>


With the above notation, if <math>\mathbf{F}</math> is any smooth vector field on <math>\R^3</math>, then<ref name="Jame">{{Cite book |last=Stewart |first=James |author-link=James Stewart (mathematician) |url={{Google books|plainurl=yes|id=btIhvKZCkTsC|page=786 }} |title=Essential calculus: early transcendentals |publisher=[[Brooks/Cole]] |year=2010 |isbn=978-0-538-49739-8 |location=Australia; United States}}</ref><ref name="bath">Robert Scheichl, [http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf lecture notes] for [[University of Bath]] mathematics course</ref><math display="block">\oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}}  = \iint_{\Sigma} \nabla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. </math>

Here, the "<math>\cdot</math>" represents the [[dot product]] in <math>\R^3</math>.

=== Special case of a more general theorem ===

Stokes' theorem can be viewed as a special case of the following identity:<ref>{{Cite journal |last=Pérez-Garrido |first=A. |date=2024-05-01 |title=Recovering seldom-used theorems of vector calculus and their application to problems of electromagnetism |journal=[[American Journal of Physics]] |volume=92 |issue=5 |pages=354–359 |arxiv=2312.17268 |doi=10.1119/5.0182191 |bibcode=2024AmJPh..92e.354P |issn=0002-9505}}</ref>
<math display="block">\oint_{\partial\Sigma} (\mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}})\,\mathbf{g}  = \iint_{\Sigma}\left[ \mathrm{d}\mathbf{\Sigma}\cdot\left( \nabla\times\mathbf{F}- \mathbf{F}\times\nabla\right)\right]\mathbf{g}, </math>
where <math>\mathbf{g}</math> is any smooth vector or scalar field in <math>\mathbb{R}^3</math>. When <math>\mathbf{g}</math> is a uniform scalar field, the standard Stokes' theorem is recovered.

== Proof ==
The proof of the theorem consists of 4 steps. We assume [[Green's theorem]], so what is of concern is how to boil down the three-dimensional complicated problem (Stokes' theorem) to a two-dimensional rudimentary problem (Green's theorem).<ref>{{Cite book |last=Colley |first=Susan Jane |author-link=Susan Jane Colley |url=https://universitytime.home.blog/wp-content/uploads/2020/04/1.vector-cal-4.pdf |title=Vector calculus |publisher=Pearson |year=2012 |isbn=978-0-321-78065-2 |edition=4th |location=Boston |pages=500–3 |oclc=732967769}}</ref>  When proving this theorem, mathematicians normally deduce it as a special case of a [[Generalized Stokes theorem|more general result]], which is stated in terms of [[differential form]]s, and proved using more sophisticated machinery. While powerful, these techniques require substantial background, so the proof below avoids them, and does not presuppose any knowledge beyond a familiarity with basic vector calculus and linear algebra.<ref name="bath" /> At the end of this section, a short alternative proof of Stokes' theorem is given, as a corollary of the generalized Stokes' theorem.

=== Elementary proof ===
==== First step of the elementary proof (parametrization of integral) ====
As in ''{{slink|#Theorem}}'', we reduce the dimension by using the natural parametrization of the surface.  Let {{math|'''''ψ'''''}} and {{mvar|γ}} be as in that section, and note that by change of variables
<math display="block">\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,\mathrm{d}\mathbf{\Gamma}}
= \oint_{\gamma}{\mathbf{F}(\boldsymbol{\psi}(\mathbf{\gamma}))\cdot\,\mathrm{d}\boldsymbol{\psi}(\mathbf{\gamma})}
= \oint_{\gamma}{\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot J_{\mathbf{y}}(\boldsymbol{\psi})\,\mathrm{d}\gamma}</math>
where {{mvar|J<sub>''y''</sub>ψ}} stands for the [[Jacobian matrix and determinant|Jacobian matrix]] of {{mvar|ψ}} at {{math|1=''y'' = ''γ''(''t'')}}.

Now let {{math|{'''e'''<sub>''u''</sub>, '''e'''<sub>''v''</sub>}<nowiki/>}} be an orthonormal basis in the coordinate directions of {{math|'''R'''<sup>2</sup>}}.<ref group="note" name="orf2">In this article,
<math display="block">\mathbf{e}_u= \begin{bmatrix}1 \\ 0 \end{bmatrix} ,
\mathbf{e}_v = \begin{bmatrix}0 \\ 1 \end{bmatrix} .</math>
Note that, in some textbooks on vector analysis, these are assigned to different things.
For example, in some text book's notation, {{math|{'''e'''<sub>''u''</sub>, '''e'''<sub>''v''</sub>}<nowiki/>}} can mean the following {{math|{'''t'''<sub>''u''</sub>, '''t'''<sub>''v''</sub>}<nowiki/>}} respectively.
In this article, however, these are two completely different things.
<math display="block">\mathbf{t}_{u} =\frac {1}{h_u}\frac {\partial \varphi}{\partial u} \, ,
\mathbf{t}_{v} =\frac {1}{h_v}\frac {\partial \varphi}{\partial v}.</math>
Here,
<math display="block">h_u = \left\|\frac {\partial \varphi}{\partial u}\right\| , 
h_v = \left\|\frac {\partial \varphi}{\partial v} \right\|, </math>
and the "<math>\| \cdot \|</math>" represents [[Euclidean distance|Euclidean norm]].
</ref>

Recognizing that the columns of {{math|''J''<sub>'''y'''</sub>'''''ψ'''''}} are precisely the partial derivatives of {{math|'''''ψ'''''}} at {{math|'''y'''}}, we can expand the previous equation in coordinates as
<math display="block">\begin{align}
\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,\mathrm{d}\mathbf{\Gamma}}
&= \oint_{\gamma}{\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot J_{\mathbf{y}}(\boldsymbol{\psi})\mathbf{e}_u(\mathbf{e}_u\cdot\,\mathrm{d}\mathbf{y}) + \mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot J_{\mathbf{y}}(\boldsymbol{\psi})\mathbf{e}_v(\mathbf{e}_v\cdot\,\mathrm{d}\mathbf{y})} \\
&=\oint_{\gamma}{\left(\left(\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot\frac{\partial\boldsymbol{\psi}}{\partial u}(\mathbf{y})\right)\mathbf{e}_u + \left(\mathbf{F}(\boldsymbol{\psi}(\mathbf{y}))\cdot\frac{\partial\boldsymbol{\psi}}{\partial v}(\mathbf{y})\right)\mathbf{e}_v\right)\cdot\,\mathrm{d}\mathbf{y}}
\end{align}</math>

==== Second step in the elementary proof (defining the pullback) ====
The previous step suggests we define the function
<math display=block>\mathbf{P}(u,v) = \left(\mathbf{F}(\boldsymbol{\psi}(u,v))\cdot\frac{\partial\boldsymbol{\psi}}{\partial u}(u,v)\right)\mathbf{e}_u + \left(\mathbf{F}(\boldsymbol{\psi}(u,v))\cdot\frac{\partial\boldsymbol{\psi}}{\partial v}(u,v) \right)\mathbf{e}_v</math>

Now, if the scalar value functions <math>P_u</math> and <math>P_v</math> are defined as follows,
<math display=block>{P_u}(u,v) = \left(\mathbf{F}(\boldsymbol{\psi}(u,v))\cdot\frac{\partial\boldsymbol{\psi}}{\partial u}(u,v)\right)</math>
<math display=block>{P_v}(u,v) =\left(\mathbf{F}(\boldsymbol{\psi}(u,v))\cdot\frac{\partial\boldsymbol{\psi}}{\partial v}(u,v) \right) </math>
then,
<math display=block>\mathbf{P}(u,v) = {P_u}(u,v) \mathbf{e}_u + {P_v}(u,v)  \mathbf{e}_v .</math>

This is the [[Pullback (differential geometry)|pullback]] of {{math|'''F'''}} along {{math|'''''ψ'''''}}, and, by the above, it satisfies
<math display=block>\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,\mathrm{d}\mathbf{l}}=\oint_{\gamma}{\mathbf{P}(\mathbf{y})\cdot\,\mathrm{d}\mathbf{l}}
=\oint_{\gamma}{( {P_u}(u,v) \mathbf{e}_u + {P_v}(u,v)  \mathbf{e}_v)\cdot\,\mathrm{d}\mathbf{l}}
</math>

We have successfully reduced one side of Stokes' theorem to a 2-dimensional formula; we now turn to the other side.

==== Third step of the elementary proof (second equation) ====
First, calculate the partial derivatives appearing in [[Green's theorem]], via the [[General Leibniz rule|product rule]]:
<math display=block>\begin{align}
\frac{\partial P_u}{\partial v} &= \frac{\partial (\mathbf{F}\circ \boldsymbol{\psi})}{\partial v}\cdot\frac{\partial \boldsymbol\psi}{\partial u} + (\mathbf{F}\circ \boldsymbol\psi) \cdot\frac{\partial^2 \boldsymbol\psi}{\partial v \, \partial u} \\[5pt]
\frac{\partial P_v}{\partial u} &= \frac{\partial (\mathbf{F}\circ \boldsymbol{\psi})}{\partial u}\cdot\frac{\partial \boldsymbol\psi}{\partial v} + (\mathbf{F}\circ \boldsymbol\psi) \cdot\frac{\partial^2 \boldsymbol\psi}{\partial u \, \partial v}
\end{align}</math>

Conveniently, the second term vanishes in the difference, by [[equality of mixed partials]].  So,<ref group="note" name="naiseki">
For all <math>\textbf{a}, \textbf{b} \in \mathbb{R}^{n}</math>, for all <math>A ; n\times n </math> [[square matrix]], <math>\textbf{a}\cdot A \textbf{b} = \textbf{a}^\mathsf{T}A \textbf{b}</math> and therefore <math>\textbf{a}\cdot A \textbf{b} = \textbf{b}  \cdot  A^\mathsf{T} \textbf{a}</math>.
</ref>
<math display="block">\begin{align}
\frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v}
&= \frac{\partial (\mathbf{F}\circ \boldsymbol\psi)}{\partial u}\cdot\frac{\partial \boldsymbol\psi}{\partial v} - \frac{\partial (\mathbf{F}\circ \boldsymbol\psi)}{\partial v}\cdot\frac{\partial \boldsymbol\psi}{\partial u} \\[5pt]
&= \frac{\partial \boldsymbol\psi}{\partial v}\cdot(J_{\boldsymbol\psi(u,v)}\mathbf{F})\frac{\partial \boldsymbol\psi}{\partial u} - \frac{\partial \boldsymbol\psi}{\partial u}\cdot(J_{\boldsymbol\psi(u,v)}\mathbf{F})\frac{\partial \boldsymbol\psi}{\partial v} && \text{(chain rule)}\\[5pt]
&= \frac{\partial \boldsymbol\psi}{\partial v}\cdot\left(J_{\boldsymbol\psi(u,v)}\mathbf{F}-{(J_{\boldsymbol\psi(u,v)}\mathbf{F})}^{\mathsf{T}}\right)\frac{\partial \boldsymbol\psi}{\partial u}
\end{align}
</math>

But now consider the matrix in that quadratic form—that is, <math>J_{\boldsymbol\psi(u,v)}\mathbf{F}-(J_{\boldsymbol\psi(u,v)}\mathbf{F})^{\mathsf{T}}</math>.  We claim this matrix in fact describes a cross product.
Here the superscript "<math> {}^{\mathsf{T}} </math>" represents the [[Transpose|transposition of matrices]].

To be precise, let <math>A=(A_{ij})_{ij}</math> be an arbitrary {{math|3 × 3}} matrix and let
<math display="block">\mathbf{a}= \begin{bmatrix}a_1 \\ a_2 \\ a_3\end{bmatrix} = \begin{bmatrix}A_{32}-A_{23} \\ A_{13}-A_{31} \\ A_{21}-A_{12}\end{bmatrix}</math>

Note that {{math|'''x''' ↦ '''a''' × '''x'''}} is linear, so it is determined by its action on basis elements.  But by direct calculation
<math display="block">\begin{align}
\left(A-A^{\mathsf{T}}\right)\mathbf{e}_1 &= \begin{bmatrix} 0 \\ a_3 \\ -a_2 \end{bmatrix} = \mathbf{a}\times\mathbf{e}_1\\
\left(A-A^{\mathsf{T}}\right)\mathbf{e}_2 &= \begin{bmatrix} -a_3 \\ 0 \\ a_1 \end{bmatrix} = \mathbf{a}\times\mathbf{e}_2\\
\left(A-A^{\mathsf{T}}\right)\mathbf{e}_3 &= \begin{bmatrix} a_2 \\ -a_1 \\ 0 \end{bmatrix} = \mathbf{a}\times\mathbf{e}_3
\end{align}</math>
Here, {{math|{'''e'''<sub>''1''</sub>, '''e'''<sub>''2''</sub>, '''e'''<sub>''3''</sub>}<nowiki/>}} represents an orthonormal basis in the coordinate directions of <math>\R^3</math>.<ref group="note" name=orf3>In this article,
<math display="block">\mathbf{e}_1= \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix} ,
\mathbf{e}_2 = \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} ,
\mathbf{e}_3 = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix} .</math>
Note that, in some textbooks on vector analysis, these are assigned to different things.</ref>

Thus {{math|1=(''A'' − ''A''{{sup|T}})'''x''' = '''a''' × '''x'''}} for any {{math|'''x'''}}.

Substituting <math>{(J_{\boldsymbol\psi(u,v)}\mathbf{F})}</math> for {{mvar|A}}, we obtain
<math display=block>\left({(J_{\boldsymbol\psi(u,v)}\mathbf{F})} - {(J_{\boldsymbol\psi(u,v)}\mathbf{F})}^{\mathsf{T}} \right) \mathbf{x} =(\nabla\times\mathbf{F})\times \mathbf{x}, \quad \text{for all}\, \mathbf{x}\in\R^{3}</math>

We can now recognize the difference of partials as a [[Triple product|(scalar) triple product]]:
<math display="block">\begin{align}
\frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v}
&= \frac{\partial \boldsymbol\psi}{\partial v}\cdot(\nabla\times\mathbf{F}) \times \frac{\partial \boldsymbol\psi}{\partial u} = (\nabla\times\mathbf{F})\cdot \frac{\partial \boldsymbol\psi}{\partial u} \times \frac{\partial \boldsymbol\psi}{\partial v}
\end{align}</math>

On the other hand, the definition of a [[surface integral]] also includes a triple product—the very same one!
<math display="block">\begin{align}
\iint_\Sigma (\nabla\times\mathbf{F})\cdot \, d\mathbf{\Sigma}
&=\iint_D {(\nabla\times\mathbf{F})(\boldsymbol\psi(u,v))\cdot\frac{\partial \boldsymbol\psi}{\partial u}(u,v)\times \frac{\partial \boldsymbol\psi}{\partial v}(u,v)\,\mathrm{d}u\,\mathrm{d}v}
\end{align}</math>

So, we obtain
<math display=block> \iint_\Sigma (\nabla\times\mathbf{F})\cdot \,\mathrm{d}\mathbf{\Sigma } = \iint_D \left( \frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v} \right) \,\mathrm{d}u\,\mathrm{d}v </math>

==== Fourth step of the elementary proof (reduction to Green's theorem) ====
Combining the second and third steps and then applying [[Green's theorem]] completes the proof.
Green's theorem asserts the following: for any region D bounded by the Jordans closed curve γ and two scalar-valued smooth functions <math>P_u(u,v), P_v(u,v)</math> defined on D;

<math display=block>\oint_{\gamma}{( {P_u}(u,v) \mathbf{e}_u + {P_v}(u,v)  \mathbf{e}_v)\cdot\,\mathrm{d}\mathbf{l}}
= \iint_D \left( \frac{\partial P_v}{\partial u} - \frac{\partial P_u}{\partial v} \right) \,\mathrm{d}u\,\mathrm{d}v </math>

We can substitute the conclusion of STEP2 into the left-hand side of Green's theorem above, and substitute the conclusion of STEP3 into the right-hand side.
[[Q.E.D.]]

=== Proof via differential forms ===
The functions <math> \R^3\to\R^3 </math> can be identified with the differential 1-forms on <math> \R^3</math> via the map
<math display="block">F_x\mathbf{e}_1+F_y\mathbf{e}_2+F_z\mathbf{e}_3 \mapsto F_x\,\mathrm{d}x + F_y\,\mathrm{d}y + F_z\,\mathrm{d}z .</math>

Write the differential 1-form associated to a function {{math|'''F'''}} as {{math|''ω''<sub>'''F'''</sub>}}.  Then one can calculate that
<math display=block>\star\omega_{\nabla\times\mathbf{F}}=\mathrm{d}\omega_{\mathbf{F}},</math>
where {{math|★}} is the [[Hodge star]] and <math>\mathrm{d}</math> is the [[exterior derivative]].  Thus, by generalized Stokes' theorem,<ref>{{Cite book |last=Edwards |first=Harold M. |url={{Google books|plainurl=yes|v9mFWa_SKGIC}} |title=Advanced calculus: a differential forms approach |publisher=Birkhäuser |year=1994 |isbn=978-0-8176-3707-1 |edition=3rd |location=Boston}}</ref>
<math display="block">\oint_{\partial\Sigma}{\mathbf{F}\cdot\,\mathrm{d}\mathbf{\gamma}}
=\oint_{\partial\Sigma}{\omega_{\mathbf{F}}}
=\int_{\Sigma}{\mathrm{d}\omega_{\mathbf{F}}}
=\int_{\Sigma}{\star\omega_{\nabla\times\mathbf{F}}}
=\iint_{\Sigma}{\nabla\times\mathbf{F}\cdot\,\mathrm{d}\mathbf{\Sigma}}
</math>

== Applications ==

=== Irrotational fields ===
In this section, we will discuss the irrotational field ([[lamellar vector field]]) based on Stokes' theorem.

'''Definition 2-1 (irrotational field).''' A smooth vector field {{math|'''F'''}} on an [[open set|open]] <math>U\subseteq\R^3</math> is ''irrotational'' ([[lamellar vector field]]) if {{math|1=∇ × '''F''' = 0}}.

This concept is very fundamental in mechanics; as we'll prove later, if  {{math|'''F'''}} is ''irrotational'' and  the domain of {{math|'''F'''}} is [[simply connected]], then {{math|'''F'''}} is a [[conservative vector field]].

==== Helmholtz's theorem ====
In this section, we will introduce a theorem that is derived from Stokes' theorem and characterizes vortex-free vector fields. In classical mechanics and fluid dynamics it is called [[Helmholtz theorem (classical mechanics)|Helmholtz's theorem]].

'''Theorem 2-1 (Helmholtz's theorem in fluid dynamics).'''<ref name="DTPO" /><ref name=fujimno/>{{rp|142}}  Let <math> U\subseteq\R^3</math> be an [[open set|open]] [[subset]] with a lamellar vector field {{math|'''F'''}} and let {{math|''c''<sub>0</sub>, ''c''<sub>1</sub>: [0, 1] → ''U''}} be piecewise smooth loops. If there is a function {{math|''H'': [0, 1] × [0, 1] → ''U''}} such that
* '''[TLH0]''' {{mvar|H}} is piecewise smooth,
* '''[TLH1]''' {{math|1=''H''(''t'', 0) = ''c''<sub>0</sub>(''t'')}} for all {{math|''t'' ∈ [0, 1]}},
* '''[TLH2]''' {{math|1=''H''(''t'', 1) = ''c''<sub>1</sub>(''t'')}} for all {{math|''t'' ∈ [0, 1]}},
* '''[TLH3]''' {{math|1=''H''(0, ''s'') = ''H''(1, ''s'')}}  for all {{math|''s'' ∈ [0, 1]}}.
Then,
<math display=block>\int_{c_0} \mathbf{F} \, \mathrm{d}c_0=\int_{c_1} \mathbf{F} \, \mathrm{d}c_1</math>

Some textbooks such as Lawrence<ref name="DTPO" /> call the relationship between {{math|''c''<sub>0</sub>}} and {{math|''c''<sub>1</sub>}} stated in theorem 2-1 as "homotopic" and the function {{math|''H'': [0, 1] × [0, 1] → ''U''}} as "homotopy between {{math|''c''<sub>0</sub>}} and {{math|''c''<sub>1</sub>}}". However, "homotopic" or "homotopy" in above-mentioned sense are different (stronger than) [[Homotopy|typical definitions]] of "homotopic" or "homotopy"; the latter omit condition [TLH3].  So from now on we refer to homotopy (homotope) in the sense of theorem 2-1 as a ''tubular homotopy (resp. tubular-homotopic)''.{{refn|group="note"|name="TLH"|There do exist textbooks that use the terms "homotopy" and "homotopic" in the sense of Theorem 2-1.<ref name="DTPO" />  Indeed, this is very convenient ''for the specific problem'' of conservative forces.  However, both uses of homotopy appear sufficiently frequently that some sort of terminology is necessary to disambiguate, and the term "tubular homotopy" adopted here serves well enough for that end.}}

===== Proof of Helmholtz's theorem =====
[[File:Domain of singular 2 cube.svg|thumb|The definitions of {{math|''γ''<sub>1</sub>, ..., ''γ''<sub>4</sub>}}]]

In what follows, we [[abuse of notation|abuse notation]] and use "<math>\oplus</math>" for concatenation of paths in the [[fundamental groupoid]] and "<math>\ominus</math>" for reversing the orientation of a path.

Let {{math|1=''D'' = [0, 1] × [0, 1]}}, and split {{math|∂''D''}} into four line segments {{math|''γ''{{sub|''j''}}}}. 
<math display=block>\begin{align}
\gamma_1:[0,1] \to D;\quad&\gamma_1(t) = (t, 0) \\
\gamma_2:[0,1] \to D;\quad&\gamma_2(s) = (1, s) \\
\gamma_3:[0,1] \to D;\quad&\gamma_3(t) = (1-t, 1) \\
\gamma_4:[0,1] \to D;\quad&\gamma_4(s) = (0, 1-s)
\end{align}</math>
so that <math display="block">\partial D = \gamma_1 \oplus \gamma_2 \oplus \gamma_3 \oplus \gamma_4</math>

By our assumption that {{math|''c''<sub>0</sub>}} and {{math|''c''<sub>1</sub>}} are piecewise smooth homotopic, there is a piecewise smooth homotopy {{math|''H'': ''D'' → ''M''}}
<math display=block>\begin{align}
\Gamma_i(t) &= H(\gamma_{i}(t)) && i=1, 2, 3, 4 \\
\Gamma(t) &= H(\gamma(t)) =(\Gamma_1 \oplus \Gamma_2 \oplus \Gamma_3 \oplus \Gamma_4)(t)
\end{align}</math>

Let {{mvar|S}} be the image of {{mvar|D}} under {{mvar|H}}.  That
<math display=block> \iint_S \nabla\times\mathbf{F}\, \mathrm{d}S = \oint_\Gamma \mathbf{F}\, \mathrm{d}\Gamma </math>
follows immediately from Stokes' theorem. {{math|'''F'''}} is lamellar, so the left side vanishes, i.e.
<math display=block>0=\oint_\Gamma \mathbf{F}\, \mathrm{d}\Gamma = \sum_{i=1}^4 \oint_{\Gamma_i} \mathbf{F} \, \mathrm{d}\Gamma </math>

As {{mvar|H}} is tubular(satisfying [TLH3]),<math>\Gamma_2 = \ominus \Gamma_4</math> and <math>\Gamma_2 = \ominus \Gamma_4</math>.  Thus the line integrals along {{math|Γ<sub>2</sub>(''s'')}} and {{math|Γ<sub>4</sub>(''s'')}} cancel, leaving
<math display=block>0=\oint_{\Gamma_1} \mathbf{F} \, \mathrm{d}\Gamma +\oint_{\Gamma_3} \mathbf{F} \, \mathrm{d}\Gamma</math>

On the other hand, {{math|1=''c''{{sub|1}} = Γ{{sub|1}}}}, <math>c_3 = \ominus \Gamma_3</math>, so that the desired equality follows almost immediately.

=== Conservative forces ===
Above Helmholtz's theorem gives an explanation as to why the work done by a conservative force in changing an object's position is path independent. First, we introduce the Lemma 2-2, which is a corollary of and a special case of Helmholtz's theorem.

'''Lemma 2-2.'''<ref name="DTPO" /><ref name=lee/> Let <math> U\subseteq\R^3</math> be an [[open set|open]] [[subset]], with a Lamellar vector field {{math|'''F'''}} and a piecewise smooth loop {{math|''c''<sub>0</sub>: [0, 1] → ''U''}}. Fix a point {{math|'''p''' ∈ ''U''}}, if there is a homotopy {{math|''H'': [0, 1] × [0, 1] → ''U''}} such that
* '''[SC0]''' {{mvar|H}} is ''piecewise smooth'',
* '''[SC1]''' {{math|1= ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'')}} for all {{math|''t'' ∈ [0, 1]}},
* '''[SC2]''' {{math|1= ''H''(''t'', 1) = '''p'''}} for all {{math|''t'' ∈ [0, 1]}},
* '''[SC3]''' {{math|1= ''H''(0, ''s'') = ''H''(1, ''s'') = '''p'''}} for all {{math|''s'' ∈ [0, 1]}}.
Then,
<math display=block>\int_{c_0} \mathbf{F} \, \mathrm{d}c_0=0</math>

Above Lemma 2-2 follows from theorem 2–1. In Lemma 2-2, the existence of {{mvar|H}} satisfying [SC0] to [SC3] is crucial;the question is whether such a homotopy can be taken for arbitrary loops. If {{mvar|U}} is simply connected, such {{mvar|H}} exists. The definition of [[simply connected space]] follows:

'''Definition 2-2 (simply connected space).'''<ref name="DTPO" /><ref name=lee/> Let <math>M\subseteq\R^n</math> be non-empty and [[Connected space#Path connectedness|path-connected]].  {{mvar|M}} is called [[simply connected]] if and only if for any continuous loop, {{math|''c'': [0, 1] → ''M''}} there exists a continuous tubular homotopy {{math|''H'': [0, 1] × [0, 1] → ''M''}} from {{mvar|c}} to a fixed point {{math|''p'' ∈ ''c''}}; that is,
* '''[SC0']''' {{mvar|H}} is ''continuous'',
* '''[SC1]''' {{math|1= ''H''(''t'', 0) = ''c''(''t'')}} for all {{math|''t'' ∈ [0, 1]}},
* '''[SC2]''' {{math|1= ''H''(''t'', 1) = '''p'''}} for all {{math|''t'' ∈ [0, 1]}},
* '''[SC3]''' {{math|1= ''H''(0, ''s'') = ''H''(1, ''s'') = '''p'''}}  for all {{math|''s'' ∈ [0, 1]}}.

The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately if the M is simply connected.　However, recall that simple-connection only guarantees the existence of a ''continuous'' homotopy satisfying [SC1-3]; we seek a piecewise smooth homotopy satisfying those conditions instead.

Fortunately, the gap in regularity is resolved by the [[Whitney's approximation theorem]].<ref name=lee/>{{rp|136,421}}<ref name="ptr">{{Cite journal |last=Pontryagin |first=L. S. |date=1959 |title=Smooth manifolds and their applications in homotopy theory |url=https://people.math.rochester.edu/faculty/doug/otherpapers/pont4.pdf |journal=American Mathematical Society Translations |series=Series 2 |language=en |location=Providence, Rhode Island |publisher=[[American Mathematical Society]] |volume=11 |pages=1–114 |doi=10.1090/trans2/011/01 |isbn=978-0-8218-1711-7 |mr=0115178 |translator-last1=Hilton |translator-first1=P. J.}} See theorems 7 & 8.</ref> In other words, the possibility of finding a continuous homotopy, but not being able to integrate over it, is actually eliminated with the benefit of higher mathematics. We thus obtain the following theorem.

'''Theorem 2-2.'''<ref name="DTPO" /><ref name=lee/> Let <math>U\subseteq\R^3</math> be [[open set|open]] and simply connected with an irrotational vector field {{math|'''F'''}}. For all piecewise smooth loops {{math|''c'': [0, 1] → ''U''}}
<math display=block>\int_{c_0} \mathbf{F} \, \mathrm{d}c_0 = 0</math>

=== Maxwell's equations ===
{{See also|Maxwell's equations#Circulation and curl}}
In the physics of [[electromagnetism]], Stokes' theorem provides the justification for the equivalence of the differential form of the [[Maxwell–Faraday equation]] and the [[Ampère's circuital law|Maxwell–Ampère equation]] and the integral form of these equations. For Faraday's law, Stokes theorem is applied to the electric field, <math>\mathbf{E}</math>:
<math display=block>\oint_{\partial\Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{\nabla}\times \mathbf{E} \cdot \mathrm{d} \mathbf{S} .</math>

For Ampère's law, Stokes' theorem is applied to the magnetic field, <math>\mathbf{B}</math>:
<math display=block>\oint_{\partial\Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{\nabla}\times \mathbf{B} \cdot \mathrm{d} \mathbf{S} .</math>

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}

[[Category:Electromagnetism]]
[[Category:Mechanics]]
[[Category:Vectors (mathematics and physics)]]
[[Category:Vector calculus]]
[[Category:Theorems in calculus]]